双缝实验单时间步状态转移矩阵列结构及目标列分配疑问

Hey there! Let's break this down step by step, since state transition matrices can feel confusing when you're first mapping them to physical systems.

First, let's lock in the core rule for this matrix: each column represents a current state of the particle, and each row represents the probability of moving to that next state from the column's current state. In other words, the value at matrix[row][col] is the probability that a particle in state col will end up in state row after one time step.

Now let's map each of the 8 columns (and rows, since they correspond to the same set of states) to physical positions in your double-slit setup:

• Column 1: Initial emission vertex (the particle's starting point)
  Looking at this column, you'll see 1/2 in row 2 and 1/2 in row 3: that's the 50/50 chance the particle travels to the left slit or right slit. The top row (row 1) is 0 because the particle can't stay at the initial vertex—100% chance it moves away, which matches your observation.
• Column 2: Left slit vertex
  The values here are 1/3 in rows 4, 5, 6: this means from the left slit, the particle has an equal 1/3 chance to move to each of three target states in one time step.
• Column 3: Right slit vertex
  Similarly, 1/3 in rows 6,7,8: from the right slit, the particle has a 1/3 chance to move to each of three target states. Notice row 6 is shared here—that's the middle target that both slits can reach!
• Columns 4-8: Target/absorbing states
  Each of these columns has a 1 in its matching row (e.g., column 4 has 1 in row 4) and 0 elsewhere. These are "terminal" states: once the particle hits a target, it stays there forever (hence the 1 probability of remaining in the same state).

Now let's tackle your specific questions:

1. Why are there so many columns?
   Each column corresponds to a distinct state the particle can occupy. When you count them up: 1 initial vertex + 2 slits + 5 target states (3 unique physical targets, with one shared between both slits, plus two exclusive to each slit) = 8 total states. Each needs its own column to track transitions from that state.
2. Why are the targets (from each slit) in different columns?
   Even if two paths lead to the same physical target (like the middle one in column 6), each target position is treated as a unique state for tracking purposes. The separate columns let us model the fact that the particle can reach that middle target from either slit, while the other targets are only reachable from one slit. Plus, since targets are absorbing states, each needs its own column to represent that once the particle is there, it doesn't move elsewhere.

Also, your note about needing two time steps to reach targets makes perfect sense: the particle starts at the initial vertex (step 0), moves to a slit (step 1), then moves to a target (step 2)—which aligns with how this matrix works (applying it twice would give the 2-step transition probabilities).

备注：内容来源于stack exchange，提问作者opmgal